Generalizations of certain vectors

I have a few questions about generalizing a few 4-vectors into tensors based on physical and intuitive arguments.

The first question I have is if I can form a Stress-Energy-Momentum tensor out of the energy-momentum wave 4-vector [tex]\hbar k_{\alpha}[/tex]?

The formation of the stress-energy tensor in GR came out by associating the energy momentum 4-vector with a 3-volume. What if I preform that same association with the wave 4-vector?

The result I obtain is still of course a stress-energy-momentum tensor when multiplied by [tex]\hbar[/tex], however by it self it is simply just a wave-tensor, with the [tex]k_{0i}[/tex] components representing momentum and energy, but with the [tex]k_{ij}[/tex] components representing the flux of [tex]k[/tex] or the [tex]\frac{dk}{dt}[/tex] across [tex]dA[/tex].

Is this result generally known and unaccepted, or is it physically absurd and hence dismissed?

The second part of my post hinges on the former idea, however I'll still write it for fun.

the wave 4-vector is commonly contracted with the coordinate 4-vector. However, if the wave tensor exists, it will need to be contracted with a coordinate tensor. using dimensional analysis the spatial components of the coordinate tensor [tex]x^{\alpha\beta}[/tex], [tex]x_{ij}[/tex], are 4-volumes, each containing the multiplication of two, 2-surfaces. one, a spatial surface, and the other a temporal surface, with the diagonal elements being a typical 4-volume from GR [tex] v=\int \sqrt{-g}d^{4}x[/tex].

Pending the first part of the post, is this an acceptable generalization of the coordinate 4-vector into a tensor? any help would be appreciated.